While I do agree that taking care of your Wyvern is part of your job as a DRG, no matter how actively you attempt to keep it alive, when an NM, that you'll be fighting 15 times an hour, such as Dragua that spams -agas and other AoEs, your Wyvern will be dead in a matter of seconds.
Here are some possible solutions:
1) Reduce Call Wyvern Timer
2) Allow Cruor Buffs to go to pets. This will not add an unreasonable boost to Healing Breaths. In Pre-Abyssea days, Wyvern Healing Breaths were much more effective, granting a much higher percentage of HP Recovery than it does now. Now, not much has changed in regard to how much HP Healing Breaths grant, making them less effective as they used to be.
3) Grant a moderate DEF bonus, and a major Magic DEF Bonus. While many AoEs in Abyssea are physical, the majority seems to be magic based. Granting a major Physical DEF boost would cause people to make their Wyvern tank.
Please do not:
Give Wyverns invincibility. This would take away a major aspect of DRG, even rendering a certain ability pointless.